Asymptotic behaviour of solutions to semi-linear elliptic equations on the half-cylinder

Various, but closely related, forms of a Phragmén-Lindelöf principle are established for certain classes of semi-linear elliptic equations (including systems) defined on a half-cylinder. For example, it is shown that under either homogeneous Dirichlet or Neumann lateral boundary conditions, the smooth solution either fails to exist globally, or when it does exist globally it must tend asymptotically to zero with increasingly large distance along the cylinder from the base. For uniformly elliptic equations, the decay to zero is at least exponential. The method employed relies upon cross-sectional estimates and is additionally applicable to both the finite and whole cylinder. In the latter case, global existence fails for all non-trivial solutions. © 1992 Birkhäuser Verlag.